Related papers: Intransitive dice tournament is not quasirandom
An edge coloring of a tournament $T$ with colors $1,2,\dots,k$ is called \it $k$-transitive \rm if the digraph $T(i)$ defined by the edges of color $i$ is transitively oriented for each $1\le i \le k$. We explore a conjecture of the second…
We determine the exact probabilities of the different isomorphism classes of tournaments that result from random sets of three and four independent dice drawn from the balanced uniform model of 3-sided dice.
Tournament solutions provide methods for selecting the "best" alternatives from a tournament and have found applications in a wide range of areas. Previous work has shown that several well-known tournament solutions almost never rule out…
A tournament is called locally transitive if the outneighbourhood and the inneighbourhood of every vertex are transitive. Equivalently, a tournament is locally transitive if it avoids the tournaments $W_4$ and $L_4$, which are the only…
Linial and Morgenstern conjectured that, among all $n$-vertex tournaments with $d\binom{n}{3}$ cycles of length three, the number of cycles of length four is asymptotically minimized by a random blow-up of a transitive tournament with all…
We consider a two player simultaneous-move game where the two players each select any permissible $n$-sided die for a fixed integer $n$. A player wins if the outcome of his roll is greater than that of his opponent. Remarkably, for $n>3$,…
We prove a conjecture dating back to a 1978 paper of D.R.\ Musser~\cite{musserirred}, namely that four random permutations in the symmetric group $\mathcal{S}_n$ generate a transitive subgroup with probability $p_n > \epsilon$ for some…
In this article, a new method for characterizing nontransitive dice is de- scribed. This new method is then used to describe the "Nontransitive Identities" (NI) that are possible for 3 dice with 3, 4 and 5 sides each as well as for 5 dice…
We consider the manipulability of tournament rules for round-robin tournaments of $n$ competitors. Specifically, $n$ competitors are competing for a prize, and a tournament rule $r$ maps the result of all $\binom{n}{2}$ pairwise matches…
A tournament organizer must select one of $n$ possible teams as the winner of a competition after observing all $\binom{n}{2}$ matches between them. The organizer would like to find a tournament rule that simultaneously satisfies the…
We study a high-dimensional analog for the notion of an acyclic (aka transitive) tournament. We give upper and lower bounds on the number of $d$-dimensional $n$-vertex acyclic tournaments. In addition, we prove that every $n$-vertex…
A central objective in Ramsey theory is determining whether restricted families of discrete structures necessarily contain substantially larger homogeneous substructures, compared to the unrestricted structures. In the setting of…
In 1970, Statistics giant, Bradley Efron, amazed the world by coming up with a set of four dice, let's call them A,B,C,D, whose faces are marked with [0,0,4,4,4,4], [3,3,3,3,3,3],[2,2,2,2,6,6],[1,1,1,5,5,5] respectively, where die A beats…
A tournament is $k$-spectrally monomorphic if all the $k\times k$ principal submatrices of its adjacency matrix have the same characteristic polynomial. Transitive $n$-tournaments are trivially $k$-spectrally monomorphic. We show that there…
An {\it inversion} of a tournament $T$ is obtained by reversing the direction of all edges with both endpoints in some set of vertices. Let ${\rm inv}_k(T)$ be the minimum length of a sequence of inversions using sets of size at most $k$…
In this paper, we give a direct construction for a set of dice realizing any given tournament $T$. The construction for a tournament with $n$ vertices requires a number of sides on the order of $n$, which appears to be the best general…
We consider the permutation analogue of Penney's game for words. Two players, in order, each choose a permutation of length $k\ge3$; then a sequence of independent random values from a continuous distribution is generated, until the…
This article deals with ranking methods. We study the situation where a tournament between $n$ players $P_1$, $P_2$, \ldots $P_n$ gives the ranking $P_1 \succ P_2 \succ \cdots \succ P_n$, but, if the results of $P_n$ are no longer taken…
We consider a generalisation of Kelly's conjecture which is due to Alspach, Mason, and Pullman from 1976. Kelly's conjecture states that every regular tournament has an edge decomposition into Hamilton cycles, and this was proved by K\"uhn…
Positions of chess players in intransitive (rock-paper-scissors) relations are considered. Namely, position A of White is preferable (it should be chosen if choice is possible) to position B of Black, position B of Black is preferable to…